1000 cc gamma pressure swing

[matt brown]

OK, I'm an alpha guy, but a recent look at gamma event sequence enticed me to look further. Here's what I arrive at for the pressure swing on an idealized gamma where the sole focus is on heating and swept volume of the displacer. I'm not going to worry about any input or regenerator mumbo-jumbo. Here goes...and Vincent, this is right up Elon's alley:

Consider a gamma with 1000cc displacer volume where the hot space has zero clearance volume TDC and the cycle is 300-600k. From basic thermal, we know (or should know) that T is linear P when Cv (constant volume heating process). Let's give this 1000cc volume a mass of gas that I'll define merely as 60m, thus 6m per 100cc.

eng A: displacer sweeps thru 500cc, so at BDC there's 500cc hot space & 500cc cold space. Since the cycle is 300-600k, the thermal ratio is 2, whereby the density in the cold space will always be 2x the density in the hot space. Thus, at BDC, there's 20m in 500cc hot space and 40m in 500cc cold space.

eng B: displacer sweeps thru 750cc, so at BDC there's 36m in 750cc hot space and 24m in 250cc cold space.

eng C: displacer sweeps thru 800cc, so at BDC there's 40m in 800cc hot space and 20m in 200cc cold space.

Sorry the values couldn't come out as an easy visual, but when corrected, the starting density with 1000cc in 300k cold space was 6m per 100cc, whereby the 300k cold space relative density eng A @ BDC = 8m per 100cc, while eng B @ BDC = 9.6m per 100cc, and eng C @ BDC = 10m per 100cc. Then if we transpose 6m per 100cc into 6 bar, we find Pmax A=8 bar, B=9.6 bar, C=10 bar.

Remember, this is idealized, and when the hot space has a real clearance volume (aka Vmin>0) then the pressure swing will be less for each example. So, even if you have an endless free source of 600k input, you'll need something the size of Kiochi's monster to generate 1-2 HP unless you drastically increase the buffer pressure beyond typical DIY range.

Someone, tell me I screwed up...

[VincentG]

Thanks for the work here Matt, I'd like to see the math behind this. Playing around with a gas law calculator it seems engine c reached an average gas temp of 500k. Working backwards that engine running at 1 bar would have a pressure gain of 10psi. Is this ballpark what you got?

How are you calculating potential hp?

[matt brown]

Consider a gamma with 1000cc displacer volume where the hot space has zero clearance volume TDC and the cycle is 300-600k. From basic thermal, we know (or should know) that T is linear P when Cv (constant volume heating process). Let's give this 1000cc volume a mass of gas that I'll define merely as 60m, thus 6m per 100cc.

eng A: displacer sweeps thru 500cc, so at BDC there's 500cc hot space & 500cc cold space. Since the cycle is 300-600k, the thermal ratio is 2, whereby the density in the cold space will always be 2x the density in the hot space. Thus, at BDC, there's 20m in 500cc hot space and 40m in 500cc cold space.

eng B: displacer sweeps thru 750cc, so at BDC there's 36m in 750cc hot space and 24m in 250cc cold space.

eng C: displacer sweeps thru 800cc, so at BDC there's 40m in 800cc hot space and 20m in 200cc cold space.

Remember, this is idealized, and when the hot space has a real clearance volume (aka Vmin>0) then the pressure swing will be less for each example. So, even if you have an endless free source of 600k input, you'll need something the size of Kiochi's monster to generate 1-2 HP unless you drastically increase the buffer pressure beyond typical DIY range.

Here this is in table format:

gamma 1000cc.png (9.4 KiB) Viewed 4674 times

Note that despite hot:cold space variations, the cold space density (m per 100cc) is always 2x the hot space density, since all 3 versions have 300k cold space and 600k hot space. The 1-2 HP claim would hinge on many typical nasties where the wildcard would likely be cycle rate (rpm). I think betas & gammas are reservoir schemes and I'll stay with alphas (when scheming SE).

[matt brown]

I decided to take this esoteric gamma a tad further by adding a 200cc power piston to eng C. Again with isolated ideal events, consider power piston at TDC with 20m 600k gas in 800cc hot space and 40m 300k gas in 200cc cold space. As piston expands 100cc from TDC, total eng volume is now 1100cc and the gas will redistribute itself thruout eng wherein at this point 30m is in 733cc hot space and 30m is in 366cc cold space. Pressure has dropped 20% thruout eng with hot space density of 4m per 100cc and cold space density of 8m per 100cc. Note that this new cold space includes the 100cc currently in power piston cylinder, so excluding this volume, the displacer cold space volume has increased from 200cc to 266cc (while the displacer hot space has decreased from 800cc to 733cc).

Now, as power piston reaches BDC, total eng vol is 1200cc, and the gas will redistribute itself thruout eng whereby at this point 30m is in 800cc hot space and 30m is in 400cc cold space (pis/cyl and displacer cold space). At power piston BDC, hot space density is 3.75m per 100cc and cold space density is 7.5m per 100cc. Oddly, as the power piston moves from TDC to BDC, the displacer requires a "bubble" to balance pressure (not kinematic friendly).

Next, when displacer moves to TDC while power piston remains at BDC, eng vol is 1200cc with 5m density thruout (all 300k). Then power piston can return (compress) to TDC and restore cycle to 1000cc with 6m density.

A sharp eye will notice a few things: (1) 200cc displacer cold space with 200cc pis/cyl vol represents 2:1 'compression' ratio (2) most expansion output is during first 1/2 of piston travel (3) even under this extreme example, the pressure swing was only 10m to 7.5m during expansion, and 5m to 6m during compression.

I'm going to look into this displacer 'bubble' more, as well as the weird 50-50 gas distribution between hot & cold spaces by 100cc piston expansion and same at 200cc piston expansion.

[matt brown]

I'm going to look into this displacer 'bubble' more...

edit: 'bubble' should be bobble (like bobblehead)

[VincentG]

Two questions come to mind. Why are you considering the compression ratio into only the cold volume? Also, I don't quite understand how the mass of the gas is all that critical when not dealing with internal combustion. Wouldn't pressure trump mass here?

And third I suppose, a well designed displacer chamber can reduce the "off" volume to a much lower percentage than 200cc.

[matt brown]

The event sequence for each engine is idealized as distinct events merely to show maximum values (PVT whatever). Forgive 'bobble' issue, a major faux pas. When I added 100cc expansion, I got 30m in 733cc hot space and 30m in 366 cold space with P in hot space equal P in cold space AND each space had equal m. Yet, hot space could easily remain 800cc with cold space of 300cc (as piston expanded), but then...

800cc hot space with ~34.4m gas total and 4.3m per 100cc
300cc cold space with ~25.8m gas total and 8.6m per 100cc

(gas totals 60.2m as a rounding error vs actual 60m total)

The point thruout this is tracking pressure via tracking gas mass within cold space since we know m density in cold space is always 2x m density in hot space due to hot space T is 2x cold space T (deg k) if 300-600k cycle. Meanwhile, tracking cold space pressure is via pressure inverse volume when constant T (aka isothermal process). The only tricky part here is that the gas will redistribute itself until pressure equalizes thruout engine. Simply saying pressure per hot and cold space is inadequate, kinda like assigning rpm to an engine.

Ideally, the gas only compresses into the cold space, so I considered the compression ratio as simply the maximum cold space divided by displacer cold space. Again, idealized distinct events, but when displacer cold space is maximum AND power piston volume is maximum, then this is the lowest temperature, pressure, and density in cycle. Therefore, Wneg (compression) will be the least here.

An ideal displacer chamber probably swings between 0 hot space and 0 cold space.

OK, I'm an alpha guy, but I got an altered state during this gamma dive. Vincent, you're getting to be a real stud at this, so do you see how to get a call from the Nobel Committee or the MacArthur Foundation ???

[VincentG]

The point thruout this is tracking pressure via tracking gas mass within cold space since we know m density in cold space is always 2x m density in hot space due to hot space T is 2x cold space T (deg k) if 300-600k cycle. Meanwhile, tracking cold space pressure is via pressure inverse volume when constant T (aka isothermal process). The only tricky part here is that the gas will redistribute itself until pressure equalizes thruout engine. Simply saying pressure per hot and cold space is inadequate, kinda like assigning rpm to an engine.

I did not mean to reduce this down to such a simple equation. But I do think that the standard math is inadequate, in that it is trying to describe a static event in a dynamic situation. While we can map PVT, I think in reality the effect has a rubber band like delay, and changing volume is not the same as just changing temperature or pressure. Each may have a distinct order of events and in a running engine, all are changing at the same time so maybe it's just impossible to quantify in an equation. That is part of what makes an alpha so hard for me to understand.

I'll have to add some smoke into the engine, but I would imagine that when the displacer is held at mid position, while the pressure does equalize and the piston(when disconnected from crank) maintains its position, the gas is constantly circulating in the displacer cylinder instead of sitting in isolated pockets. So even here in what seems a static snapshot in time, it is still a dynamic cycle.

OK, I'm an alpha guy, but I got an altered state during this gamma dive. Vincent, you're getting to be a real stud at this, so do you see how to get a call from the Nobel Committee or the MacArthur Foundation ???

Matt your gonna inflate my ego here, if only I had the time that would grant me. But just for hoots it's 845-394-1825

[matt brown]

Koichi Hirata FPSE.png (67.13 KiB) Viewed 4572 times

Despite the free piston, this Kiochi diagram remains the typical Stirling cycle sequence which I'll refer to as the closed cycle version. If we consider D=displacer and P=power piston, then...

closed cycle:

1-2 D up (heating)
2-3 P up (expansion)
3-4 D down (cooling)
4-1 P down (compression)

Note that this event sequence represents a typical 4 process cycle where processes are interleaved in an alternating pattern (nothing new here). However, this scheme can be altered into an open cycle by reversing the order of the last P & D events and adding 1 or 2 valves...

open cycle:

1-2 D up (heating)
2-3 P up (expansion)

3-2 P down (exhaust via adding a port near BDC in power cylinder)
2-1 D down (intake via adding a snifter valve in cold plate)

Note that in this event sequence, the processes are not interleaved. The closed cycle version is up-up-down-down while the open cycle version is more akin up-down-up-down. Assuming both open & closed versions are the same LTD type SE with 1 bar buffer pressure, then both will require the same input to drive the power piston to BDC. Considered as ideal isolated events, both will require the displacer to reach the cold plate (Pmax) prior power piston moving, but the open cycle should run faster since venting some gas after expansion (open cycle) will lower the pressure faster than cooling the gas (closed cycle). Once we achieve a valid open cycle SE scheme, everything changes...

[VincentG]

How do you define valid?

[matt brown]

How do you define valid?

How about valid meaning logical basis.

Note that in the above closed cycle, the event sequence favors continuous motion...like typical 90 deg out-of-phase.

Meanwhile in above open cycle, the event sequence favors discontinuous motion...like 180 deg dwell for displacer during piston motion then 180 deg dwell for piston during displacer motion. If both cycles lack regen then both are Otto, but no (or less) regen for open cycle would be easier to achieve per previous post (P drop trumps T drop).

The neat thing is finding a simple way to achieve an open cycle opens up a wide range of possibilities. This walkthru was via a typical LTD gamma, but not restricted to LTD. With a little imagination, a whole new scheme evolves where the 180 deg dwell of both displacer and power piston are easily incorporated into a sweet scheme. Here's the clue: I'm gonna buy 2 LTD to play with...

[matt brown]

I've been reviewing the Kiochi LTD monster lately and this guy is clever !!! I still need to check mass flow for his 1.6 compression ratio vs 1.33 thermal ratio (300-400k) but I suspect his 1 bar buffer pressure equal ambient pressure is more than coincidental. IOW not just to compensate leakage (he used air) since I found some fascinating parallels to my tracking mass flow early in this thread.

[VincentG]

I'd like to hear the what and why on dwelling the power piston for that long.

The piston port at bdc seems to introduce some constant volume expansion. I wonder if the intake port could be tuned to add some constant volume compression via tunnel ram effect just before the displacer moves gas to the hot side.

Best thing about open cycle is it eliminates gassing out.

[matt brown]

If we consider D=displacer and P=power piston, then...

closed cycle:

1-2 D up (heating)
2-3 P up (expansion)
3-4 D down (cooling)
4-1 P down (compression)

open cycle:

1-2 D up (heating)
2-3 P up (expansion)

3-2 P down (exhaust via adding a port near BDC in power cylinder)
2-1 D down (intake via adding a snifter valve in cold plate)

Once we achieve a valid open cycle SE scheme, everything changes...

Both above cycles have the same displacer and piston motion, but where the first half of both cycles have the same 2 events vs the second half of both cycles have 2 slightly different 'events'. The trick here is isolating each event before combining them into a 'cycle'.

Looking at the typical closed cycle event sequence, if we consider the time of each processes relative to 360 deg motion, we find that 1 cycle requires 720 deg of time when ideal isolated events where

1-2 D up (heating) = 180 deg
2-3 P up (expansion) = 180 deg
3-4 D down (cooling) = 180 deg
4-1 P down (compression) = 180 deg

includes an alternating dwell for both displacer and piston wherein both displacer and piston dwell 1/2 the cycle time.
The closed cycle manages 1 cycle in 360 deg of time via overlapping these events in classic out-of-phase motion. So, a clean closed cycle requires 4 dwell periods which total 1/2 the cycle time.

Meanwhile, looking at the open cycle event sequence, the cycle timing is very similar and 1 cycle requires 720 deg of time when ideal isolated events where

1-2 D up (heating) = 180 deg
2-3 P up (expansion) = 180 deg

3-2 P down (exhaust) = 180 deg
2-1 D down (intake) = 180 deg

Similar the closed cycle, the open cycle requires dwell for 1/2 the total cycle time, but consists of 2 longer dwells (equal 4 shorter dwells of closed cycle). I'm still pondering effects of this open cycle scheme reduced to typical 90 deg out-of-phase motion...

[matt brown]

I'd like to hear the what and why on dwelling the power piston for that long.

The piston port at bdc seems to introduce some constant volume expansion. I wonder if the intake port could be tuned to add some constant volume compression via tunnel ram effect just before the displacer moves gas to the hot side.

Best thing about open cycle is it eliminates gassing out.

Hold on...no such thing as 'constant volume' compression or expansion.

The basic scheme remains similar closed cycle where

(1) displacer moves cold gas to hot space
(2) gas expands power piston

but typical closed cycle gamma has

(3) displacer moves hot gas to cold space
(4) power piston compresses gas

vs open cycle gamma has

(3) gas expands power piston then exhausts 'some' gas to ambient
(4) displacer moves hot gas to cold space and intakes 'some' ambient gas

A sharp eye will question this open cycle scheme and note that if gas pressure in power piston at BDC is equal ambient pressure then there's no 'exhaust' to ambient. OK, no natural exhaust via dP if equal, but this volume must be removed so power piston can return to 'TDC'. Then when displacer moves hot gas to cold space, there is less gas in engine and pressure drop is compensated by admitting air thru valve in cold plate. Note 2 things: (1) all gas in power cylinder must be removed before displacer moves, so simple port in power piston at BDC is inadequate (2) allowing minimal effort to exhaust power cylinder gas after expansion (ya know, pressure on both sides of piston being equal) nixes typical closed cycle 'compression' after displacer moves gas to cold space. A single valve in the cold plate would suffice, but probably require mechanical actuation at power piston BDC (and cold space clearance volume).

Nearly obvious, this open cycle scheme bars LTD vacuum scheme where pressure in engine after expansion is less than ambient